Thermodynamics and Chemistry

CHAPTER 9 MIXTURES

9.7 ACTIVITY OF ANUNCHARGEDSPECIES 273

reference state based on mole fraction or molality, this process brings the system to the
reference state of componentiat pressurep^0. The change ofiin this case is given by
integration of diDVidp:

refi .p^0 /iD

Zp 0

p

Vidp (9.7.14)

The appropriate partial molar volumeViis the molar volumeViorVAof the pure sub-
stance, or the partial molar volumeVB^1 of solute B at infinite dilution.
Suppose we want to use a reference state for solute B based on concentration. Because
the isothermal pressure change involves a small change of volume,cBchanges slightly
during the process, so that the right side of Eq.9.7.14is not quite the correct expression for
refc;B.p^0 /c;B.

We can derive a rigorous expression forrefc;B.p^0 /c;Bas follows. Consider an ideal-

dilute solution of solute B at an arbitrary pressurep, with solute chemical potential

given byBDrefc;BCRTln.cB=c/(Table9.2). From this equation we obtain



@B

@p



T;fnig

D

@refc;B

@p

!

T

CRT



@ln.cB=c/

@p



T;fnig

(9.7.15)

The partial derivative.@B=@p/T;fnigis equal to the partial molar volumeVB(Eq.

9.2.49), which in the ideal-dilute solution has its infinite-dilution valueVB^1. We

rewrite the second partial derivative on the right side of Eq.9.7.15as follows:



@ln.cB=c/

@p



T;fnig

D

1

cB



@cB

@p



T;fnig

D

1

nB=V



@.nB=V /

@p



T;fnig

DV



@.1=V /

@p



T;fnig

D

1

V



@V

@p



T;fnig

DT (9.7.16)

HereTis the isothermal compressibility of the solution, which at infinite dilution is

T^1 , the isothermal compressibility of the pure solvent. Equation9.7.15becomes

VB^1 D

@refc;B

@p

!

T

CRT T^1 (9.7.17)

Solving for drefc;Bat constantT, and integrating fromptop^0 , we obtain finally

refc;B.p^0 /c;BD

Zp 0

p

VB^1 RT T^1

dp (9.7.18)

We are now able to write explicit formulas forifor each kind of mixture component.
They are collected in Table9.6on the next page.
Considering a constituent of a condensed-phase mixture, by how much is the pressure
factor likely to differ from unity? If we use the valuespD 1 bar andT D 300 K, and
assume the molar volume of pureiisViD 100 cm^3 mol^1 at all pressures, we find that
iis0:996in the limit of zero pressure, unity at 1 bar,1:004at 2 bar,1:04at 10 bar, and